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The Hopf algebra of rooted trees, free Lie algebras, and Lie series. (English) Zbl 1116.17004
The author develops a new approach which translates in the language of labeled rooted trees the definition and the main properties of arbitrary Hall bases of free Lie algebras. He creates a rewriting algorithm using labeled rooted trees in the dual Poincaré-Birkhoff-Witt (PBW) basis associated to an arbitrary Hall set, that aims handling Lie series, exponentials of Lie series, and related series written in the PBW basis. This allows performing computations related to the Baker-Campbell-Haussdorff (BCH) formula and its generalizations in an arbitrary Hall basis. In particular, he establishes explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. The author shows that his approach is actually based on an explicit description of an epimorphism of Hopf algebras (and its kernel) from the commutative Hopf algebra of labeled rooted trees to the shuffle Hopf algebra. The motivation of the research is not only purely algebraic and combinatorial. It involves applications to numerical methods for solving systems of ordinary differential equations, in nonlinear control theory, etc.

17B01 Identities, free Lie (super)algebras
05C05 Trees
05C85 Graph algorithms (graph-theoretic aspects)
05E99 Algebraic combinatorics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
65L99 Numerical methods for ordinary differential equations
93B25 Algebraic methods
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